matrix diagonalisation and calcul

Question

Hello guys can someone help me with that please ? And thanks for that

$A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 1 \\ -2 & 0 & -1 \end{bmatrix}, \tag 1$

and we ask us to diagonalize and after that calculate $\A^n$.

So for the diagonalisation i found that:

So can someone help me to calculate from that $\A^n$ please and thanks.

Answer 1

$A^2=P^{-1}DPP^{-1}DP=P^{-1}D^2P$

Similarly, $A^n=P^{-1}D^nP$ by induction.

It's easy to compute the powers of a diagonal matrix.

Answer 2

$A=PDP^{-1}$. Therefore, $A^{2}=(PDP^{-1})(PDP^{-1})=PD(P^{-1}P)DP^{-1}=PD^{2}P^{-1}$. By induction, $A^{n}=PD^{n}P^{-1}$. Since $D$ is a diagonal matrix, $D^{n}$ can be found by raising the diagonal elements to their $n$th power.